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In figure, PA and PB are tangents to the circle with center O such that $∠ APB = 50 °$ . Write the measure of $∠ O A B .$

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90 o

50 o

60 o

25 o

answer is A.

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Detailed Solution

Given that PA and PB are tangents to the circle with center O such that

∠ APB = 50 °

.
We have to find

∠ O A B .

The tangent of a circle at any point in a circle is perpendicular to the radius through the point of contact
The required figure geometry is shown below,
Question Image

Since the length of the tangent from an external point to a circle is equal. So,

P A = P B

. In triangle

P B A, ∠ P B A = ∠ P A B

. Because angles opposite the equal sides are equal. The sum of the three angles of a triangle is equal to 180 degrees. So,

∠ A P B + ∠ P B A + ∠ P A B = 180 ° ⇒ 50 ° + ∠ P A B + ∠ P A B = 180 ° ⇒ 2 ∠ P A B = 180 ° − 50 ° ⇒ ∠ P A B = 65 °

From the theorem, the tangent of a circle at any point in a circle is perpendicular to the radius through the point of contact. So,

O A ⊥ P A

and

∠ O A P = 90 °

Thus,

∠ P A B + ∠ O A B = 90 ° ⇒ 65 ° + ∠ O A B = 90 ° ⇒ ∠ O A B = 90 ° − 65 ° ⇒ ∠ O A B = 25 °

The value of

∠ O A B

25 °

Therefore, the correct option is 1.

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